Peaktopeak analysis is a relatively simple method to assess the capacity utilization of an industry over time. An advantage of peaktopeak analysis is that it requires information on only one output measure and one input measure, and hence is suited to estimating capacity utilization with only Level 1 data (see Table A.1). Peaktopeak analysis has been applied in fisheries by Ballard and Roberts (1977), Ballard and Blomo (1978) and Hsu (2003).
Peaktopeak analysis is based on an underlying assumption that output is a function of the level of inputs and a technology trend, such that
(1)
where Y_{t} is the output in time, t; a_{0} is a proportionality constant; V_{t} is a composite or aggregate index of inputs; and T_{t} is the technology trend that represents productivity change. An implicit assumption in the use of a composite index of inputs is that the technology displays constant returns to scale. That is, increasing all inputs will result in a proportional increase in output.
The level of technology is determined by the average rate of change in productivity between peak years, where productivity is given by Y_{t}/V_{t} (i.e. average output per unit of input). The technology in any one year is thus
(2)
where m is the length of time from the previous peak year, and n is the length of time to the following peak year, and T_{tm} is the level of technology at the previous peak (i.e. year m) equivalent to the average productivity (e.g. catch per unit of effort) in that period. The other term on the right hand side (i.e. the term inside the brackets) represents the cumulative change in productivity between the two peaks. This is added to the average productivity in the previous peak year (i.e. year m) to give an estimate of the average productivity of capacity in subsequent years.
An alternative way of estimating the level of technology between peaks is given by
(3)
where Y_{n}/V_{n} is the average productivity in the upper peak and Y_{m}/V_{m} is the average productivity in the lower peak. The term in the brackets represents the average change in productivity between the two peaks. Both approaches produce identical results.
Assuming the proportionality constant has a value of 1, the estimate of the level of technology is equivalent to the capacity level of productivity (i.e. T_{t} = Y_{t}*/V_{t}, where Y_{t}* is the capacity level of output). From this, the capacity level of production can be estimated from the product of the inputs and the capacity level of productivity, such that
Y_{t}* = V_{t}T_{t} (4)
and capacity utilization can be estimated by
CU_{t} = Y_{t}*/Y_{t}. (5)
A particular difficulty in interpreting the results of a peaktopeak analysis in fisheries is that no consideration is given to changes in the stock level. Apparent changes in productivity may be due to either changes in technology (the underlying assumption of the technique) or changes in the stock level.
This problem may be particularly pertinent in developing fisheries, where catch rates may increase rapidly initially, with the main peak occurring in the middle of the time series. Subsequent declines in catch rates may reflect falling stock levels. However, if the main peak is used as the last peak in the series (all other years showing a steady decline), it is likely that the technique will overestimate capacity output and underestimate capacity utilization.
The problem can be minimized by including lower peaks rather than successively higher peaks as is generally used in other industries that do not rely upon a biological resource base.
Data on the artisanal fishing sector in Nigeria were used as an example of how peaktopeak analysis can be used to estimate capacity. The data were derived from Amire (2003), and are presented in Table B.1.
Table B.1  Nigerian artisanal fisheries productivity, 19761994




Average catch per: 

Year 
Canoes 
Fishers 
Production 
Canoe 
Fisher 
1976 
134 337 
413 832 
327 561 
2 438 
0.792 
1977 
137 447 
424 838 
331 280 
2 410 
0.780 
1978 
138 447 
425 298 
336 138 
2 431 
0.790 
1979 
133 728 
446 152 
356 888 
2 669 
0.800 
1980 
133 723 
459 065 
274 158 
2 050 
0.597 
1981 
120 142 
440 592 
323 916 
2 696 
0.735 
1982 
105 239 
416 959 
377 683 
3 589 
0.906 
1983 
129 555 
472 122 
376 984 
2 910 
0.798 
1984 
109 638 
342 219 
246 784 
2 251 
0.721 
1985 
80 688 
302 234 
140 873 
1 746 
0.466 
1986 
77 134 
408 927 
160 169 
2 077 
0.392 
1987 
76 644 
437 465 
145 755 
1 902 
0.333 
1988 
77 144 
447 850 
185 181 
2 400 
0.413 
1989 
77 155 
470 250 
171 332 
2 221 
0.364 
1990 
76 981 
452 187 
170 459 
2 214 
0.377 
1991 
77 093 
457 102 
168 211 
2.182 
0.368 
1992 
77 076 
459 847 
184 407 
2 393 
0.401 
1993 
77 050 
456 381 
106 276 
1 379 
0.233 
1994 
77 073 
457 775 
124 117 
1 610 
0.271 
Source: Amire (2003).
The choice of input may have an impact on the measure of capacity output and, consequently, capacity utilization. In the Nigerian artisanal fleet, the number of canoes active in the fishery had declined over time while the number of fishers remained relatively constant (a result of more fishers operating per canoe). Over the same period, motorization increased in the fishery from 8.7 percent in 1996 to 20.8 percent in 1994 (Amire, 2003). As a result, it would be expected that there was substantial technological change in the fishery. Developing a composite index of inputs in such a case is difficult without first estimating a production function and imposing constant returns to scale.
For purposes of illustration, capacity was assessed using both canoes and fishers (separately) for the input measure. From Table B.1, it can be seen that the peak productivity periods for both inputs were 1976, 1979, 1982, 1988 and 1992. These peaks also are apparent by graphing the catch per unit input series (Figure B.1).
Figure B.1  Catchperunit input, Nigerian artisanal fleet
Table B.2  Peaktopeak analysis using canoes as input measure
Year (t) 
Canoes (V_{t}) 
Production (Y_{t}) 
CPUE (Y_{t}/V_{t}) 
Average technological change^{a} 
Capacity CPUE (T_{t}) 
Capacity output (Y_{t}*) 
Utilization rate (Y_{t}/Y_{t}*) 
1976 
134 337 
327 561 
2 438 
 
2 438 
327 561 
100% 
1977 
137 447 
331 280 
2 410 
0.0768 
2 515 
345 701 
96% 
1978 
138 247 
336 138 
2 431 
0.0768 
2 592 
358 330 
94% 
1979 
133 728 
356 888 
2 669 
0.0768 
2 669 
356 888 
100% 
1980 
133 723 
274 158 
2 050 
0.3067 
2 975 
397 885 
69% 
1981 
120 142 
323 916 
2 696 
0.3067 
3 282 
394 321 
82% 
1982 
105 239 
377 683 
3 589 
0.3067 
3 589 
377 683 
100% 
1983 
129 555 
376 984 
2 910 
0.1981 
3 391 
439 289 
86% 
1984 
109 638 
246 784 
2 251 
0.1981 
3 193 
350 041 
71% 
1985 
80 688 
140 873 
1 746 
0.1981 
2 995 
241 631 
58% 
1986 
77 134 
160 169 
2 077 
0.1981 
2 797 
215 711 
15% 
1987 
76 644 
145 755 
1 902 
0.1981 
2 599 
199 161 
73% 
1988 
77 144 
185 181 
2 400 
0.1981 
2 400 
185 181 
100% 
1989 
77 155 
171 332 
2 221 
0.0020 
2 398 
185 055 
93% 
1990 
76 981 
170 459 
2 214 
0.0020 
2 396 
184 485 
92% 
1991 
77 093 
168 211 
2 182 
0.0020 
2 395 
184 600 
91% 
1992 
77 076 
184 407 
2 393 
0.0020 
2 393 
184 407 
100% 
1993 
77 050 
106 276 
1 379 
0.0020 
2 391 
184 192 
58% 
1994 
77 073 
124 117 
1 610 
0.0020 
2 389 
184 094 
67% 
Note: Peak years in bold, a) estimated by [(Y_{n}/V_{n})(Y_{m}/V_{m})]/(nm)
The analyses, undertaken in an Excel spreadsheet, are given in Tables B.2 and B.3 using canoes and fisher numbers respectively. Average technological change was estimated between the peak years (indicated in bold). For example, between 1976 and 1979, average productivity change was (2.6692.438)/(41) = 0.0768.
Capacity CPUE is estimated by adding the average technological change to the preceding year’s value. Capacity output is estimated by multiplying the capacity CPUE by the input level. The utilization rate is estimated by dividing actual output by capacity output.
Table B.3  Peaktopeak analysis using number of fishers as input measure
Year (t) 
Fishers (V_{t}) 
Production (Y_{t}) 
CPUE (Y_{t}/V_{t}) 
Average technological change^{a} 
Capacity CPUE (T_{t}) 
Capacity output (Y_{t}*) 
Utilization rate (Y_{t}/Y_{t}*) 
1976 
413 832 
327 561 
0.792 

0.792 
327 561 
100% 
1977 
424 838 
33 1280 
0.780 
0.003 
0.794 
337 461 
98% 
1978 
425 298 
336 138 
0.790 
0.003 
0.797 
339 016 
99% 
1979 
446 152 
356 888 
0.800 
0.003 
0.800 
356 888 
100% 
1980 
459 065 
274 158 
0.597 
0.035 
0.835 
383 419 
72% 
1981 
440 592 
323 916 
0.735 
0.035 
0.871 
383 540 
84% 
1982 
416 959 
377 683 
0.906 
0.035 
0.906 
377 683 
100% 
1983 
472 122 
376 984 
0.798 
0.082 
0.824 
388 911 
97% 
1984 
342 219 
246 784 
0.721 
0.082 
0.742 
253 823 
97% 
1985 
302 234 
140 873 
0.466 
0.082 
0.660 
199 368 
71% 
1986 
408 927 
160 169 
0.392 
0.082 
0.578 
236 194 
68% 
1987 
437 465 
145 755 
0.333 
0.082 
0.496 
216 782 
67% 
1988 
447 850 
185 181 
0.413 
0.082 
0.413 
185 181 
100% 
1989 
470 250 
171 332 
0.364 
0.003 
0.410 
192 977 
89% 
1990 
452 187 
170 459 
0.377 
0.003 
0.407 
184 155 
93% 
1991 
457 102 
168 211 
0.368 
0.003 
0.404 
184 731 
91% 
1992 
459 847 
184 407 
0.401 
0.003 
0.401 
184 407 
100% 
1993 
456 381 
106 276 
0.233 
0.003 
0.398 
181 594 
59% 
1994 
457 775 
124 117 
0.271 
0.003 
0.395 
180 722 
69% 
Note: Peak years in bold, a) estimated by [(Y_{n}/V_{n})(Y_{m}/V_{m})]/(nm)
Despite differences in the input measure used, the estimated capacity output was fairly similar in both instances (Figure B.2a). The estimated capacity utilization in each year was also relatively similar (Figure B.2b).
Figure B.2 a) estimated capacity and b) estimated capacity utilization